Question: $ -3.\overline{37} \div 0.\overline{8} = {?} $
Solution: First convert the repeating decimals to fractions. $\begin{align*} 100x &= -337.3738...\\ x &= -3.3738...\end{align*} $ $\begin{align*} 99x &= -334 \\ x &= -\dfrac{334}{99}\end{align*} $ $\begin{align*} 10y &= 8.8888...\\ y &= 0.8888...\end{align*} $ $\begin{align*} 9y &= 8 \\ y &= \dfrac{8}{9}\end{align*} $ So, the problem becomes: $ -\dfrac{334}{99} \div \dfrac{8}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ -\dfrac{334}{99} \times \dfrac{9}{8} = {?} $ $ \phantom{-\dfrac{334}{99} \times \dfrac{8}{9}} = \dfrac{-334 \times 9}{99 \times 8} $ $ \phantom{-\dfrac{334}{99} \times \dfrac{8}{9}} = \dfrac{-334 \times \cancel{9}} {\cancel{99}11 \times 8} $ $ \phantom{-\dfrac{334}{99} \times \dfrac{8}{9}} = -\dfrac{334}{88} $ Simplify: ${= -\dfrac{167}{44}}$